Selection principles in function spaces with the compact-open topology

Abstract

For a Tychonoff space X, we denote by C k (X) the space of all real-valued continuous functions on X with the compact-open topology. A subset A ⊂ X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A. In this paper, the following properties for C k (X) are considered. (Formula Presented) For example, a space C k (X) satisfies S 1 (S, D) (resp., S f in (S, D)) if whenever (S n : n ∈ N) is a sequence of sequentially⋃ dense subsets of C k (X), one can take points f n ∈ S n (resp., finite F n ⊂ S n ) such that {f n : n ∈ N} (resp., {F n : n ∈ N}) is dense in C k (X). Other properties are defined similarly. In [22], we obtained characterizations these selection properties for C p (X). In this paper, we give characterizations for C k (X). © 2018, University of Nis. All rights reserved.